An operator approach to the Madelung-Bohm continuity equation

We solve the probability continuity equation within the Madelung-Bohm framework, assuming a separable phase expressed as $S(x,t) = Q(x)\dot{\nu}(t) + \mu(t)$. Using operator methods, we reformulate the wave function's amplitude into a form that is more practical for application to any given initial condition. This results in the function $F(x,t)$, which characterizes the wavefunction's amplitude, with $F$ being the transformation of the position operator via a squeeze-like operator. To demonstrate how this result can be applied, we examine two scenarios: one where the external potential is zero, causing the dynamics to originate only from the Bohm potential, and another where the form of $F$ mirrors the characteristics of wave propagation within optical waveguide arrays. In the second scenario, it is notable that the amplitude, phase, and potentials might become complex yet still comply with the Schr\"odinger equation.